Those drivers installed well (from what I understand, no errors or anything), Windows recognizes them, I have a drop-down in audio options for the card's drivers and for the onboard drivers so everything appears to check out.

However, I can't get any sound to play through the card using the drivers :/ Any solutions to this?

ThatÂ´s strange.
I take it you have already selected it as your default playback in audio options and your media player( eg.click top left in winamp/options/output/nulsoftdirect/configure)
also checked all cables,front port etc

Can you try the card in another slot/PC to verify it works ?
Lastly you could try as Crag suggested uninstall and boot in safe mode , run driver sweeper, reboot and install again.